4. Example `10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120`
Solve Equations 10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120 using Relaxation method
Solution:
Total Equations are `3`
`10x-2y-3z=205`
`-2x+10y-2z=154`
`-2x-y+10z=120`
The residuals from equations, we get
`R_1=205-10x+2y+3z`
`R_2=154+2x-10y+2z`
`R_3=120+2x+y-10z`
The table for operation is
| `R_1` | `R_2` | `R_3` | | `deltax` | -10 | 2 | 2 | | `deltay` | 2 | -10 | 1 | | `deltaz` | 3 | 2 | -10 |
Solution steps are
`1^(st)` Approximation
`R_1=205-0=205`
`R_2=154-0=154`
`R_3=120-0=120`
Maximum is `R_1=205`
`deltax=205/10=20.5`
`2^(nd)` Approximation
`R_1=205-10*20.5=205-205=0`
`R_2=154-(-2)*20.5=154--41=195`
`R_3=120-(-2)*20.5=120--41=161`
Maximum is `R_2=195`
`deltay=195/10=19.5`
`3^(rd)` Approximation
`R_1=0-(-2)*19.5=0--39=39`
`R_2=195-10*19.5=195-195=0`
`R_3=161-(-1)*19.5=161--19.5=180.5`
Maximum is `R_3=180.5`
`deltaz=180.5/10=18.05`
`4^(th)` Approximation
`R_1=39-(-3)*18.05=39--54.15=93.15`
`R_2=0-(-2)*18.05=0--36.1=36.1`
`R_3=180.5-10*18.05=180.5-180.5=0`
Maximum is `R_1=93.15`
`deltax=93.15/10=9.315`
`5^(th)` Approximation
`R_1=93.15-10*9.315=93.15-93.15=0`
`R_2=36.1-(-2)*9.315=36.1--18.63=54.73`
`R_3=0-(-2)*9.315=0--18.63=18.63`
Maximum is `R_2=54.73`
`deltay=54.73/10=5.473`
`6^(th)` Approximation
`R_1=0-(-2)*5.473=0--10.946=10.946`
`R_2=54.73-10*5.473=54.73-54.73=0`
`R_3=18.63-(-1)*5.473=18.63--5.473=24.103`
Maximum is `R_3=24.103`
`deltaz=24.103/10=2.4103`
`7^(th)` Approximation
`R_1=10.946-(-3)*2.4103=10.946--7.2309=18.1769`
`R_2=0-(-2)*2.4103=0--4.8206=4.8206`
`R_3=24.103-10*2.4103=24.103-24.103=0`
Maximum is `R_1=18.1769`
`deltax=18.1769/10=1.8177`
`8^(th)` Approximation
`R_1=18.1769-10*1.8177=18.1769-18.1769=0`
`R_2=4.8206-(-2)*1.8177=4.8206--3.6354=8.456`
`R_3=0-(-2)*1.8177=0--3.6354=3.6354`
Maximum is `R_2=8.456`
`deltay=8.456/10=0.8456`
`9^(th)` Approximation
`R_1=0-(-2)*0.8456=0--1.6912=1.6912`
`R_2=8.456-10*0.8456=8.456-8.456=0`
`R_3=3.6354-(-1)*0.8456=3.6354--0.8456=4.481`
Maximum is `R_3=4.481`
`deltaz=4.481/10=0.4481`
`10^(th)` Approximation
`R_1=1.6912-(-3)*0.4481=1.6912--1.3443=3.0355`
`R_2=0-(-2)*0.4481=0--0.8962=0.8962`
`R_3=4.481-10*0.4481=4.481-4.481=0`
Maximum is `R_1=3.0355`
`deltax=3.0355/10=0.3035`
`11^(th)` Approximation
`R_1=3.0355-10*0.3035=3.0355-3.0355=0`
`R_2=0.8962-(-2)*0.3035=0.8962--0.6071=1.5033`
`R_3=0-(-2)*0.3035=0--0.6071=0.6071`
Maximum is `R_2=1.5033`
`deltay=1.5033/10=0.1503`
`12^(th)` Approximation
`R_1=0-(-2)*0.1503=0--0.3007=0.3007`
`R_2=1.5033-10*0.1503=1.5033-1.5033=0`
`R_3=0.6071-(-1)*0.1503=0.6071--0.1503=0.7574`
Maximum is `R_3=0.7574`
`deltaz=0.7574/10=0.0757`
`13^(th)` Approximation
`R_1=0.3007-(-3)*0.0757=0.3007--0.2272=0.5279`
`R_2=0-(-2)*0.0757=0--0.1515=0.1515`
`R_3=0.7574-10*0.0757=0.7574-0.7574=0`
Maximum is `R_1=0.5279`
`deltax=0.5279/10=0.0528`
`14^(th)` Approximation
`R_1=0.5279-10*0.0528=0.5279-0.5279=0`
`R_2=0.1515-(-2)*0.0528=0.1515--0.1056=0.2571`
`R_3=0-(-2)*0.0528=0--0.1056=0.1056`
Maximum is `R_2=0.2571`
`deltay=0.2571/10=0.0257`
`15^(th)` Approximation
`R_1=0-(-2)*0.0257=0--0.0514=0.0514`
`R_2=0.2571-10*0.0257=0.2571-0.2571=0`
`R_3=0.1056-(-1)*0.0257=0.1056--0.0257=0.1313`
Maximum is `R_3=0.1313`
`deltaz=0.1313/10=0.0131`
`16^(th)` Approximation
`R_1=0.0514-(-3)*0.0131=0.0514--0.0394=0.0908`
`R_2=0-(-2)*0.0131=0--0.0263=0.0263`
`R_3=0.1313-10*0.0131=0.1313-0.1313=0`
Maximum is `R_1=0.0908`
`deltax=0.0908/10=0.0091`
`17^(th)` Approximation
`R_1=0.0908-10*0.0091=0.0908-0.0908=0`
`R_2=0.0263-(-2)*0.0091=0.0263--0.0182=0.0444`
`R_3=0-(-2)*0.0091=0--0.0182=0.0182`
Maximum is `R_2=0.0444`
`deltay=0.0444/10=0.0044`
`18^(th)` Approximation
`R_1=0-(-2)*0.0044=0--0.0089=0.0089`
`R_2=0.0444-10*0.0044=0.0444-0.0444=0`
`R_3=0.0182-(-1)*0.0044=0.0182--0.0044=0.0226`
Maximum is `R_3=0.0226`
`deltaz=0.0226/10=0.0023`
`19^(th)` Approximation
`R_1=0.0089-(-3)*0.0023=0.0089--0.0068=0.0157`
`R_2=0-(-2)*0.0023=0--0.0045=0.0045`
`R_3=0.0226-10*0.0023=0.0226-0.0226=0`
Maximum is `R_1=0.0157`
`deltax=0.0157/10=0.0016`
`20^(th)` Approximation
`R_1=0.0157-10*0.0016=0.0157-0.0157=0`
`R_2=0.0045-(-2)*0.0016=0.0045--0.0031=0.0077`
`R_3=0-(-2)*0.0016=0--0.0031=0.0031`
Maximum is `R_2=0.0077`
`deltay=0.0077/10=0.0008`
Solution By Relaxation Method.
`x=sum deltax=31.9997~=32`
`y=sum deltay=25.9998~=26`
`z=sum deltaz=20.9995~=21`
Iterations are tabulated as below
| Iteration | Operation | `deltax`
(10) | `deltay`
(10) | `deltaz`
(10) | `R_1` | `R_2` | `R_3` | | 1 | `x=y=z=0` | 0 | 0 | 0 | 205 | 154 | 120 | | 2 | `deltax=205/10=20.5` | 20.5 | 0 | 0 | 0 | 195 | 161 | | 3 | `deltay=195/10=19.5` | 0 | 19.5 | 0 | 39 | 0 | 180.5 | | 4 | `deltaz=180.5/10=18.05` | 0 | 0 | 18.05 | 93.15 | 36.1 | 0 | | 5 | `deltax=93.15/10=9.315` | 9.315 | 0 | 0 | 0 | 54.73 | 18.63 | | 6 | `deltay=54.73/10=5.473` | 0 | 5.473 | 0 | 10.946 | 0 | 24.103 | | 7 | `deltaz=24.103/10=2.4103` | 0 | 0 | 2.4103 | 18.1769 | 4.8206 | 0 | | 8 | `deltax=18.1769/10=1.8177` | 1.8177 | 0 | 0 | 0 | 8.456 | 3.6354 | | 9 | `deltay=8.456/10=0.8456` | 0 | 0.8456 | 0 | 1.6912 | 0 | 4.481 | | 10 | `deltaz=4.481/10=0.4481` | 0 | 0 | 0.4481 | 3.0355 | 0.8962 | 0 | | 11 | `deltax=3.0355/10=0.3035` | 0.3035 | 0 | 0 | 0 | 1.5033 | 0.6071 | | 12 | `deltay=1.5033/10=0.1503` | 0 | 0.1503 | 0 | 0.3007 | 0 | 0.7574 | | 13 | `deltaz=0.7574/10=0.0757` | 0 | 0 | 0.0757 | 0.5279 | 0.1515 | 0 | | 14 | `deltax=0.5279/10=0.0528` | 0.0528 | 0 | 0 | 0 | 0.2571 | 0.1056 | | 15 | `deltay=0.2571/10=0.0257` | 0 | 0.0257 | 0 | 0.0514 | 0 | 0.1313 | | 16 | `deltaz=0.1313/10=0.0131` | 0 | 0 | 0.0131 | 0.0908 | 0.0263 | 0 | | 17 | `deltax=0.0908/10=0.0091` | 0.0091 | 0 | 0 | 0 | 0.0444 | 0.0182 | | 18 | `deltay=0.0444/10=0.0044` | 0 | 0.0044 | 0 | 0.0089 | 0 | 0.0226 | | 19 | `deltaz=0.0226/10=0.0023` | 0 | 0 | 0.0023 | 0.0157 | 0.0045 | 0 | | 20 | `deltax=0.0157/10=0.0016` | 0.0016 | 0 | 0 | 0 | 0.0077 | 0.0031 | | Total | 31.9997 | 25.9998 | 20.9995 | | | |
This material is intended as a summary. Use your textbook for detail explanation.
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